HELPSubstitution and Integral by Parts

[moumouer]

I'm having big trouble when trying to figure this integral out. Please help!

Integral (from 0 to infinity): ((x^2)/a)*e^[(-x^2)/2a] dx a is a constantThanks in advance!

 

[Mark44]

I would start with u = x and dv = (1/a)x*e^-x2/(2a)dx.

 

[moumouer]

OHHHH! Thank you sooo much! I'v been thinking my head off by trying to do substitution!

And I have one more question, I'm required to calculate E(x2) too, so now instead of (X^2)/a for the first part, it becomes (x^3)/a, and the 2nd part remains the same. How do I approach this one?

Thank you very much!

 

[moumouer]

Oh wait a minute, after doing the parts, since V = - (e-x2/(2a)), now applying the fomula, uv - integral v du, how do I solve Integral - (e-x2/(2a)) dx ?

 

[moumouer]

Hi Mark44, I think your method works perfectly w/ my 2nd question. I worked it out already. But I still don't know how to solve the 1st question.

 

[Mark44]

Then I think you're stuck, unless there is some additional information we haven't seen yet. e^-x2 doesn't have a nice neat antiderivative.

 

[moumouer]

The question says Let X1...Xn be a random sample from a Rayleigh distribution with pdf
f(x) = (x/a)*e-x2/(2a), for x>0. That's it! And I'm suppose to find E(x).

 

[Dick]

The integral is from x=0 to x=infinity. It's a Gaussian integral. http://en.wikipedia.org/wiki/Gaussian_integral

 

[moumouer]

thank you so much!

 

[moumouer]

Hi Dick, I read through the article on wiki, and I'm wondering how the parameter a in my problem affect the integral. I cannot ignore a here since the reason I'm calculating E(x) is becoz I'm trying to find the Method of Moment estimator of a.

Thank you!

 

[Dick]

Set u^2=x^2/(2a). So u=x/sqrt(2a). If you write the integral in terms of u, you should be able to collect all of the a's outside of the integral.

 

[moumouer]

But it still doesn't solve the integral. it became: Integral 2u^2*e^(-u2)sqrt(2a)du

 

[Dick]

Isn't it 2*sqrt(2a)*integral u^2*e^(-u^2)*du? The u integral is just a constant. I thought you were going to use integration by parts like Mark44 suggested to reduce it to the Gaussian integral?

 

[moumouer]

Thank you very much!